Problem: For a certain hyperbola
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\]where $a > b,$ the angle between the asymptotes is $60^\circ.$  Find $\frac{a}{b}.$
Answer: We know that the point $(a,b)$ lies on an asymptote, as shown below.

[asy]
unitsize(0.8 cm);

real upperhyper(real x) {
  return (sqrt(x^2/3 - 1));
}

real lowerhyper(real x) {
  return (-sqrt(x^2/3 - 1));
}

draw(graph(upperhyper,-5,-sqrt(3) - 0.01)--(-sqrt(3),0),red);
draw(graph(lowerhyper,-5,-sqrt(3) - 0.01)--(-sqrt(3),0),red);
draw((sqrt(3),0)--graph(upperhyper,sqrt(3) + 0.01,5),red);
draw((sqrt(3),0)--graph(lowerhyper,sqrt(3) + 0.01,5),red);
draw((-5,0)--(5,0));
draw((0,-5/sqrt(3))--(0,5/sqrt(3)));
draw((-5,-5/sqrt(3))--(5,5/sqrt(3)),dashed);
draw((-5,5/sqrt(3))--(5,-5/sqrt(3)),dashed);
draw((sqrt(3),1)--(sqrt(3),0));

label("$a$", (sqrt(3)/2,0), S);
label("$b$", (sqrt(3),1/2), E, UnFill);

dot("$(a,b)$", (sqrt(3),1), NW);
[/asy]

Since the angle between the asymptotes is $60^\circ,$ $a$ is the long leg of a $30^\circ$-$60^\circ$-$90^\circ$ triangle, and $b$ is the short leg.  Thus, $\frac{a}{b} = \boxed{\sqrt{3}}.$